To solve the equation $\sqrt{3x - 3} = \sqrt{2x + 8}$ for $x$, let's follow these steps:

Step 1: Square Both Sides

Since both sides are square roots, we can eliminate the square roots by squaring both sides of the equation: $(\sqrt{3x - 3})^2 = (\sqrt{2x + 8})^2$ This simplifies to: $3x - 3 = 2x + 8$

Step 2: Solve for $x$

Now, isolate $x$ by moving terms involving $x$ to one side and constants to the other: $3x - 2x = 8 + 3$ $x = 11$

Step 3: Verify the Solution

To ensure that $x = 11$ is a valid solution, we substitute it back into the original equation: $\sqrt{3(11) - 3} = \sqrt{2(11) + 8}$ Calculating inside the square roots: $\sqrt{33 - 3} = \sqrt{22 + 8}$ $\sqrt{30} = \sqrt{30}$ Since both sides are equal, $x = 11$ is indeed a solution.

Conclusion

The correct answer is: $x = 11$

Would you like a detailed explanation of each step, or do you have any questions about the solution?

Here are 5 related questions for further practice:

1. How would you solve an equation with different square root terms, like $\sqrt{5x + 2} = \sqrt{3x + 14}$?
2. What happens if you square both sides of an equation but one side has a negative expression?
3. How do you solve an equation with a square root on one side and a linear term on the other, like $\sqrt{4x - 1} = x + 2$?
4. Can you find the solution to $\sqrt{x + 5} = 3$?
5. What are some potential pitfalls when squaring both sides of an equation?

Tip: Always verify solutions when squaring both sides, as this process can sometimes introduce extraneous solutions.